How Number of Limbs Relates to Robots and Organisms

This weekend was the weekend over which DARPA hosted its large robotics challenge where semi-autonomous robots had to perform a series of tasks simulating a disaster relief scenario. Specifically, robots had to be able to open doors, shut off water valves, drill holes in walls, climb stairs and more. It was quite the spectacle to watch and while it was impressive to see how far robotics has come, it also served as a reminder of how far robotics still has yet to go. The robots were very slow at performing their tasks and there was plenty of falling over and unintended failures. Half the robots I observed couldn't even complete the first task of opening a normal door. In short, we can all rest easy knowing that these robots aren't going to be stealing our jobs or planning a violent uprising any time soon.

One other thing which the average observer may have noticed is that the four legged robots performed significantly better than the two legged humanoid robots. To many the reasons why may seem obvious, but to others this outcome may not be so obvious. Even to those who easily made sense of this outcome, they probably have not realized how these results are consistent with the biological organisms which inhabit our planet. With that in mind, I'd like to take a moment to write about locomotion and how it relates to both robots and biological organisms alike.

Locomotion in Biology

Take a moment to recall every organism you can which moves over land using limbs. Now think about how many limbs each of these organisms have which are primarily used for moving. Now think about the number of limbs again, but this time pay special attention to the size of the organism relative to its number of limbs. Do this long enough, and a pattern should start to emerge. Specifically, smaller organisms like bugs tend to have six or more limbs while larger organisms like mammals have four and humans have only two. Once again, we're only considering the limbs used primarily for moving.

Is there something special about two limbs verses four limbs verses six limbs with regards to locomotion? As it turns out, there is. Specifically, the ability for the organism to remain stable while moving or while stationary changes as its number of limbs change.

Physics of Stability

But first, a bit of physics review. What does it mean for an organism which is standing on limbs to be stable or unstable? First, consider the limbs which are in contact with the ground. Now connect these limbs with imaginary lines to form an imaginary polygon on the ground beneath the organism. In order for an organism to be stable, its center of gravity must be somewhere within this polygon. If the center of gravity leaves this area, the organism will start to tip over and without corrective action, it will eventually topple over. So how does this relate back to having two, four, or six limbs?

Six Limbed Organisms

First, consider the six limbed bug. When the bug is stationary, it is standing on all six limbs and stable. How about when it wants to move? To move, the bug can pick up, for example, it's front right, back right, and middle left limbs. This leaves its front left, back left, and middle right limbs on still on the ground, forming a tripod. As long as the bug keeps its center of gravity near the center of its body, this is a stable position. This means the bug is now free to move its lifted limbs forward and place them down. Placing them down creates another tripod which, in turn, stabilizes the bug so it can lift up and move the limbs which were previously grounded. By using this "alternating tripod" locomotion, the bug can move while always being stable. In other words, if I had a magic wand which could freeze the bug, I could freeze the bug at any point during its locomotion and it would not fall over. So to summarize, with six limbs an organism can be stable while stationary or while in motion.

Four Limbed Organisms

How about the four limbed organism? When its stationary it has four limbs on the ground so again no problem with stability. But what about when it moves? Now there's a problem. With only four limbs, it cannot use the alternating tripod motion described earlier. It can still remain stable while in motion, but the motion would have to be very awkward. Specifically, the organism would have to shift its center of gravity to be within the right triangle formed by three of its limbs, move the now freed limb forward, and then shift its center of gravity over to the newly formed right triangle to free up another limb. This is possible, but now the organism is intentionally shifting its center of gravity as it moves which is something the six limbed organism didn't have to concern itself with. So to summarize, the four limbed organism is stable while stationary and potentially stable while moving, but requires a more complicated locomotion.

Two Limbed Organisms

Which leaves us with the two limbed organism. This organism isn't very stable while standing or moving. The problem is that two limbs only forms a line, not a polygon. So if each limb contacts the ground as only a single point, the organism will constantly be in a state of falling no matter where it shifts its center of gravity. To compensate for this, two limbed organisms have feet. Since the feet contact the ground at multiple points and not just one, the line now becomes a very narrow rectangle and stability can be achieved while standing. However, this rectangle is very narrow compared to the rectangles formed by the stationary four or six limbed organism. Consequently, if someone were to push each organism, the two limbed organism is much easier to knock over than the others. To state it more scientifically, while the four and six limbed organisms are in stable equilibrium when stationary, the two limbed organism is in unstable equilibrium. When disturbed by an outside force, the four and six limbed organisms tend to remain standing while the two limbed organism tends to start falling over.

Similarly to the four limbed organism, while it is possible for the two limbed organism to move without sacrificing stability, doing so results in a very awkward locomotion. The two limbed organism has to shift its center of gravity to be balanced on one foot so it can lift and move the other foot, and then its center of gravity has to be carefully shifted towards the newly placed foot, making sure that the center of gravity doesn't shift outside the narrow rectangle representing its stability. You can imagine how awkward and difficulty this would be.

Complexity Verses Efficiency

So if having less than six limbs makes it difficult for an organism in motion to remain stable, why do any organisms have less than six limbs? As it turns out, being unstable isn't always a bad thing. If an organism is always stable, it has to do all its own work in order to move itself. On the other hand, if an organism is unstable, it can take advantage of gravity to do some of the work of moving for it.

Consider once again the two limbed organism. Specifically, consider a human since that's the two limbed organism we're all most familiar with. Again, I'm only considering limbs used for locomotion. Our primary form of locomotion is called walking, but if you think about it, walking is really nothing more than controlled falling. To walk forward, a person lifts their foot, shifts their center of gravity forward, and then catches themselves with their lifted foot as they fall. The forward momentum from that fall then allows the person to lift their other foot and catch themselves as they fall forward again. After the first step, a portion of the forward energy is being generated by momentum as a result of gravity and this is a portion of energy which no longer needs to be generated by the organism. The result is a much more efficient locomotion than if the organism had to generate all the energy itself.

The same notion applies to running for two and four limbed organisms. Once the organism is running, the energy needed to continue running is simply the energy needed to lift itself a few inches off the ground and then to catch itself when gravity pulls it back down. While the organism is in the part of its stride where all of its limbs are off the ground, it is using no energy while still moving forward. To summarize, instability allows for greater efficiency.

Conclusion

The takeaway from all this is that when it comes to locomotion, the number of limbs a system has results in a trade off between complexity and efficiency. With more limbs, the system can be simpler because its more stable. On the other hand, with less limbs the system can be more efficient, but it also becomes more complex as it has to now be aware of its momentum and center of gravity. This could be an explanation as to why simpler organisms like bugs tend to have many limbs while more complex organisms like mammals tend to have fewer.

So what does all this have to do with the DARPA robotics challenge? What I hope I demonstrated with this rant on locomotion is that it shouldn't come as a surprise that the four limbed robots out performed the two limbed robots in this competition. For the engineers designing and building these robots, making a robot which is able to move on only two limbs adds a layer of complexity which the four limb robots don't have to worry about. However, as the field of robotics advances, two limbed robots will ultimately be superior to four limbed robots on land, especially as efficiency becomes the leading concern.